Optimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek

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1 Optmlt of Strteges for Collpsg Expe Rom Vrles Smple Rom Smple E Stek troucto We revew the propertes of prectors of ler comtos of rom vrles se o rom vrles su-spce of the orgl rom vrles prtculr, we ttempt to use theorem out optmlt to evelop the theor for choosg suspce tht results optml prectors sc Structure Cotext Smple Rom Smplg We represet the vector of rom vrle represetg expe rom permutto moel correspog to smple rom smplg s Y Y Y Y ) where Ys Ys Ys Y s) We c express Y YA + Y where Y A AY Y Y for some A ote tht E Y) vr Y) P where DPD DD ) J For exmple, settg J A, J J Y Y+ Y Ths c lso e expresse s Y Y + P ) Y where P J Y ) Y ote tht Uss s elemets of Y re of the form Y, whch re the rom vrles commol use sttstcl moels Whe prectg trget rom vrle correspog to posto, we hve use the rom vrles Y ste of Y wthout justfg tht the resultg prector s optml Y Y Also, ote tht for fxe mtrces W, X, Q, Recll tht ) the set of solutos to the equto QW X s gve + ) QW Q X Q Q Z, where Q s specfe g-verse of Q Z s rtrr mtrx Let us tke ), ote tht ) ) A Wth these expressos, Y Y + Z Ce7oc 4/4/ 7:59 AM

2 Suppose ow tht we efe ler comto of the rom vrles gve P g Y Ths ler comto s gve P g Y + g Z, where Z s rtrr vector Suppose lso tht g e s Wth ths efto, P Y + s ) e Z Y + s e Z Ths rom vrle s fucto of rtrr vector, Z, hece hs rtrr terpretto ote tht wth ths efto of g, the ler fucto s the prmeter for ut s Altertvel, we c set g e so tht the ler comto correspos to the rom vrle correspog to posto the permutto Wth ths efto, P e ) + ) Y e Z ey Thus, the rtrr fucto s zero for such ler comtos H ote tht Y D ) vec U ) t s possle to efe C C such tht C Cvec U) C vec ) U re o-stochstc, H s orthogol to ech We c lso efe C, wrte t explctl As result, we c express wrte HY Y G G) GHY+ GKwhere the seco term s costt, there re K ) rows HY We c the retur to the chrcterzto of P tht we cosere ove, see f the rtrr prt s utomtcll zero Ths strteg wll remove costrts from the vector Y A Outle of the Prolem We plce the prolem slghtl more geerl cotext Assume tht there s terest ler comto of the rom vrles efe P g Y, where g e to, ssume tht gy A P Ths ssumpto mples tht gy Let us represet suset of the rom vrles Y tht wll e oserve s Y KY where K ) ) smlr mer, let us efe Y K YA K AY Y KY Furthermore, let us efe the set Ce7oc 4/4/ 7:59 AM

3 { : s vector of costts} C LY L Furthermore, let us efe the set C LY : L s vector of costts { } { LY : L s vector of costts} C C { LY : L s vector of costts} ow suppose tht LY s optml ler use prector of P elogg to the clss C Also, let us ssume tht LY s optml ler use prector of P elogg to the clss C Fll, let MY e ler use estmtor of We woul lke to show tht LY s equvlet to the optml prector of P elogg to the clss C Moreover, we woul lke to evelop crter for efg A so tht optml prector of P elogg to the clss C s equvlet to the optml prector of P elogg to the clss C Assumptos Solutos Of relevce here s Theorem from Ro ellhouse 978), whch we restte here * Let C eote clss of pm-use estmtors of Em Y ), e, ) Let C e the correspog clss of pm-use estmtors, e, of zero, e, E e E e ) Theorem A estmtor e C s optml for f ol f for ever estmtor e of zero elogg to C t s true tht E e ) e p7) Usg the prevous otto, we ssume tht gy A P gy Let e LY LKAY e use prector of P Ths mples tht E e P), LK g AE Y Sce E Y), ths mples tht J LK g A Susttutg the expresso for A, the J LK g or equvletl, tht LK g Ths wll e true f ol f LK g Whe mplg tht ) ) ) use costrt mples tht ) ) Ce7oc 4/4/ 7:59 AM

4 g e, LK Also, sce K ), the use costrt mples tht L ext, let us ssume tht e MY s use estmtor of zero, mplg tht MK E Y ) We smplf ths expresso mkg use of J E Y), resultg J MK Usg the expresso for K ), ths J mples tht M Let us express ths s J M ow M s row vector Let us efe M m m m) where ms ms The the use costrt ms s wll e met for f ol f for ll s,,, ms m, where m Ths wll e true ol f ms km for ll s,, The theorem sttes tht the prector e LY wll e optml f ol f E e ) e or E LY ) P YM Expg terms, E LY ) P YM LK g ) A E YY ) K M E vr + E E LK g A Y, ow usg the fct tht YY ) Y) Y) Y ) ) E ) the E LY ) P YM LK g ) A vr Y) K M or E LY ) P YM LK g ) A P) K M For smple rom smplg, terms ths expresso re gve J K ) ), A, DPD P Usg these expressos, Ce7oc 4/4/ 7:59 AM 4

5 J A P) K ) ) D P D P P ) J DPDP P ) Thus, A P) K P DP P Ths expresso c e ) smplfe further Frst, DP D PD D The P P D P PD, PD σ thus P D P P D σ D ) D or P D P P D σ D D J As result, D J A P) K P ) E LY ) P YM LK g) A P) KM ) ) D LK g P ) M ) Ce7oc 4/4/ 7:59 AM 5

6 We express ths s E LY P) YM ) D ) LK g P σ ) M ) Ths expresso must e zero for the prector se o Y to e optml ote tht the clss of prectors tht we coser re prectg posto, re of the form g e For prectors of ths form, LK g ) P c e smplfe to ) LK g P ) L e ) P L P ) ep ow ep e P so tht ) ) P ) As result, ) ) L LK g P L e + ) L e ) Usg ths expresso, E LY P) YM ) ) ) D σ L ) ) M ) D e ) ) M ) D D L M e ) ) M ) We coser ths expresso more etl For ths expresso to equl zero, we, we requre D ) ) D σ L M e ) ) M ) Ce7oc 4/4/ 7:59 AM 6

7 Whe >, e, hece for ths expresso to hol, we must e le to ) ) show tht D L M Ths wll e true f ) L D ) L ) M σ M or ) L D ) L ) M σ M ) TO HERE 4/4/ Se otes from ckel Doksum, p ): Let ll possle permuttos of the elemets the populto efe the smple spce whch we efe s Ω { ω} where ω s specfc permutto of the elemets of ) A rom vrle X s fucto from Ω to such tht the set { ω: X ω) } X ) s θ for ever Β meg tht the sets re mesurele) {Wht s met θ??} * * Exmple: Suppose X ω ) u Let us ssume g, where s s )) g e For exmple, suppose let g ) The set of ω s the permuttos ω : X, X, X, X, X, X The set {,, } The possle vlues of X ω ) re Ce7oc 4/4/ 7:59 AM 7